Solving Mathematical Problems – Part 1

When I was younger, and still at school – there used to be a day I used to dread. Well, not the whole day, but some hours. OK – less than that. A minute or two, actually.

The first time I saw the math question paper in front of me.

All the problems, the numbers, and the geometric figures began swimming in front of my eyes. I’m sure some of you have also had the feeling.

It’s only when I grew older and actually began to understand the subject – I realized that there was nothing to worry about. Mathematics is not really all that hard – as long as you knew the basics of the subject. If you really, really know the basic principles, and the techniques that you can apply, then it’s just a matter of using them in the right fashion to solve any problem.

More importantly, no problem is too small, and you can always learn something new from any problem. Even failures in solving a problem can teach you something new. All you need is a curious spirit and a willingness to try. As the great 20th century mathematician George Polya said “Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery”. He goes on to add “Such experiences at a susceptible age may create a taste for mental work and leave their imprint on the mind and character for a lifetime”.

And you, my young friends are of such a age – when you are learning new things, and are full of curiosity about the world around you. If you were to instill this “taste for mental work” at this age, then it will be a lifelong companion for you, and you will have succeeded in effectively using the only organ that separates us from the other animals that inhabit our world – our brains and our capacity for thinking.

Isn’t that something worth striving for?

And you know, even at your age, that the only way to learn how to solve problems is, well, by actually solving them. Try and solve as many of them as you can. Practice, practice, practice. Sites like www.edugain.com are good sources of problems that challenge your mind. And while solving each problem, try and think about the various ways in which you could solve it (where possible, of course).

With this series of articles, we’ll try and look at various problems and techniques on solving problems in mathematics.

So what’s the very first thing you need to keep in mind when you see a problem in front of you? I’m not sure if many of you have heard of a set of books called “The Hitchhiker’s guide to the galaxy”. You’re all probably a little too young to have read them (but I recommend you do try them when you are a teenager and can appreciate them), but there’s this catchphrase in that book that applies to this situation. It’s just two words

DON’T PANIC

Yes – Don’t Panic. It’s just a problem, and no matter how complex it looks like, it is solvable using stuff you already know. Just take a deep breath, and relax.

Now, step 2. Read the question carefully. Sift out the relevant facts and numbers. The reason it probably looks complex is because it’s a long problem (not always true – some of the toughest problems are very easy to state – we’ll talk about Goldbach’s conjecture in some later articles when talking about unsolved problems).

Anyway, back to solving our problem. Pull out the relevant information and relations. See what the problem statement actually relates to among the various subjects you have learnt. In most cases, this will be straightforward, but in some cases this may be tricky. This will especially be true when the questions are of the Higher-Order-Thinking variety.

Let’s start with a very simple problem (this problem can also be found on http://www.edugain.com website).

Q: Sulekha has 8 pair of red socks and 9 pair of brown socks. All the socks are in a bag, if Sulekha picks the socks without looking at the bag, how many socks does she have to remove from the bag before she can be sure that she has a pair of a single colour?

Now, this may seem simple to some of you, but there are a lot of students who get a little confused when they see this question. So how should they approach this problem?

First, though the question tells a story about Sulekha and her rather odd collection of socks, remember that those are not important points.

The key point is that there are 8 pairs of type A, and 9 pairs of type B in a set/group.

But let’s look at the technique to solve it in terms of the stated problem.

Sulekha’s requirement is to get one pair of single colour, so she would stop once she gets either two red socks, or two brown socks.

Let’s use some simple logic to see what the possibilities are. For ease of simplicity, consider the symbols R = Red and B = Brown,

Now if she takes only one sock from the bag, she could get either (R) or (B). In this case, she definitely cannot have two of the same colour, as she took only one sock (Simple logic – you need two for a pair)

If she takes two socks out of the bag, she could get any of the following combination after she picks up both the socks - (RR) or (BB) or (BR). Note the third combination. This is the possibility of one RED and one BROWN. Since we cannot predict what she will get, so this is also not enough. Getting two socks does not guarantee there are one Black and one Red socks.

However, if she takes three socks, these are the four possibilities after all 3 socks are removed, (RRR), (BBB), (RRB) or (BBR). Note that the order of the socks does not matter, i.e. RRB is same as RBRor BRR. Here you can see that in all possibilities she is getting one pair of same colour socks. Even if one of the socks in the triplet is a different colour, atleast two of them will have the same colour. Problem solved. She just need to pick three socks to be sure that she has got at least one pair single colour.

In subsequent articles we will look at different techniques of solving problems.

In the meantime, maybe you can think about this related problem:

Sanjana has 4 pair of white socks and 9 pair of black socks. All the socks are in a bag, if Sanjana picks the socks without looking at them, she has to remove ____ socks from the bag before she can be sure that she has a pair of white color.